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QUANTUM SPACETIME AND ALGEBRAIC QUANTUM FIELD THEORY 5 1 Dorothea Bahns1, Sergio Doplicher2, Gerardo Morsella3, and 0 Gherardo Piacitelli4 2 n 1Mathematisches Institut and Courant Research Centre “Higher Order Structures a in Mathematics”, Universität Göttingen, J Bunsenstr. 3-5, D-37073 Göttingen (Germany), e-mail: [email protected] 4 2Dipartimento di Matematica, Università di Roma “La Sapienza”, 1 p.le Aldo Moro, 5, I-00185 Roma (Italy), e-mail: [email protected] ] 3Dipartimento di Matematica, Università di Roma Tor Vergata, h v.le della Ricerca Scientifica, 1, I-00133 Roma (Italy), t - e-mail: [email protected] p e 4SISSA, Via Bonomea 265, I-34136 Trieste (Italy), e-mail: [email protected] h [ 1 January 15, 2015 v 8 9 2 Abstract 3 0 Wereviewtheinvestigationsonthequantumstructureofspactime, . tobefoundatthePlanckscaleifonetakesintoaccounttheoperational 1 limitations to localization of events which result from the concurrence 0 5 ofQuantumMechanicsandGeneralRelativity. Wealsodiscussthedif- 1 ferent approaches to (perturbative) Quantum Field Theory on Quan- : tum Spacetime, and some of the possible cosmological consequences. v i X r Contents a 1 Quantum nature of spacetime at the Planck scale: why and how 2 2 The basic model: an example of Quantum Geometry 17 2.1 The basic model and its covariant representations . . . . . . . 17 2.2 Uncertainty relations and optimal localisation . . . . . . . . . 20 2.3 The C*-algebra of the basic model . . . . . . . . . . . . . . . 22 1 2.4 Many events and the diagonal map . . . . . . . . . . . . . . . 24 2.5 Planckian bounds on geometric operators . . . . . . . . . . . 27 3 Quantum Field Theory on Quantum Spacetime: the various approaches and their problems 29 3.1 Free fields and “local algebras” on QST . . . . . . . . . . . . 29 3.2 Perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Interaction terms . . . . . . . . . . . . . . . . . . . . . . . . . 32 4 Quantum Spacetime and Cosmology 37 4.1 Beyond Minkowski: a dynamical Quantum Spacetime scenario 37 4.2 Localisation on a spherically symmetric spacetime . . . . . . 38 4.3 Backreaction on Quantum Spacetime and the horizon problem 42 4.4 Further possible cosmological applications . . . . . . . . . . . 45 1 Quantum nature of spacetime at the Planck scale: why and how According to Classical General Relativity, at large scales spacetime is a pseudo Riemanniann manifold locally modelled on Minkowski space. But the concurrence with the principles of Quantum Mechanics renders this picture untenable in the small. Those theories are often reported as hardly reconcilable, but they do meet at least in a single partial principle, the Principle of Gravitational Stability against localisation of events formulated in [1, 2]: The gravitational field generated by the concentration of energy required by the Heisenberg Uncertainty Principle to localise an event in spacetime should not be so strong to hide the event itself to any distant observer - distant compared to the Planck scale. The effect of this principle is best seen considering first the effect of an observation which locates an event, say, in a spherically symmetric way aroundtheorigininspacewithaccuracya;accordingtoHeisenbergprinciple an uncontrollable energy E of order 1/a has to be transferred, which will generate a gravitational field with Schwarzschild radius R (cid:39) E (in universal units where (cid:126) = c = G = 1). Hence we must have that a (cid:38) R (cid:39) 1/a; so that a (cid:38) 1, i.e. in CGS units a (cid:38) λ (cid:39) 1.6·10−33cm. (1.1) P 2 This folklore argument is certainly very old, but its elaborations in two significant directions are surprisingly recent. First, if we consider generic uncertainties, the argument above suggests that they ought to be limited by uncertainty relations. Indeed, if we measure one of the space coordinates of our event with great precision a, but allow large uncertainties L in the knowledge of the other coordinates, the energy 1/a may spread over a thin disk of radius L and thus generate a gravitational potential that would vanish everywhere as L → ∞ (provided a, as small as we like but non zero, remains constant). This is shown by trivial computation of the Newtonian potential generated by the corresponding mass distribution; whenever such a potential is nearly vanishing, nobody would expect large General Relativistic or Quan- tum Gravitational corrections; so we can rely on that estimate. An equally elementary computation would show that the same conclu- sion holds if two space coordinates are measured with small but fixed precision a and the third one with an uncertainty L, and L → ∞. Second, if we consider the energy content of a generic quantum state where the location measurement is performed, the bounds on the uncertainties should depend also upon that energy content [3, 4] . To see this point, just suppose that our background state describes the spherically symmetric distribution of the total energy E within a sphere of radius R, with E < R. If we localise, in a spherically symmetric way, an event at the origin with space accuracy a, due to the Heisenberg Principle the total energy will be of the order 1/a+E. We must then have 1 +E < R, a otherwiseoureventwillbehiddentoanobserverlocatedfaraway,outofthe sphere of radius R around the origin. Thus, if R−E is much smaller than 1, the “minimal distance” will be much larger than 1. But if a is anyway larger than R the condition implies rather 1 +E < a. a Thus, if R−E is very small compared to 1 and R is much larger than 1, a cannot be essentially smaller than R. Nowthecausalrelationsbetweeneventsshouldalsobreakdownatscales which are so small that events cannot be localised that sharply; hence we have to expect that scale to express the range of propagation of acausal effects. 3 This naive picture suggests that, due to the principle of Gravitational Stability, initially all points of the Universe should have been causally connected. Thus we can expect that Quantum Spacetime (QST) solves the horizon problem (cf. [3] for hints in that direction, [4] or Section 4.3 below for an indication that a Quantum Spacetime with a constant Planck length should generate dynamically a range of propagation of acausal effects which solves the horizon problem). Wecomebacktothegeneraldiscussion. IfweaimatamergeofQuantum Mechanics and General Relativity we should reason in terms of concepts which are physically legitimate from the general relativistic point of view as well. One might doubt from the start about concepts like local energy and coordinates to which the Heisenberg Principle refers. Concerning the use of coordinates, one should better talk of measure- ments conditioned to the measurement of a finite number of auxiliary local quantities; in some appropriate limit, in Minkowski space, that auxiliary measurement should become the specification of a frame. Thus the use of coordinates should be legitimate at a semiclassical level. Another important reason to work with coordinates is that we are interested in the tangent space at a point equipped with normal coordinates, describing a free falling system in Einstein’s lift. Or a system in a constant gravitational field; for the outside distribution of matter on the large scale, such as the structure of the Virgo supercluster of galaxies to which we be- long, ought to have no influence on a high energy collision in the CERN collider; even if we were so clever to detect (quantum) effects of the gravitational forces between the colliding particles. Thus in a first stage it is legitimate, and physically reasonable, to study the small scale structure of Minkowski space. The spacetime symmetries of our space ought to be described by the classical Poincaré group: for the globalmotionsofourspaceshouldlookthesameinthelargeastheydointhe small, and, in the large, they should be precisely the classical symmetries. One other remark in order here concerns the very nature of the coordinates. In the Quantum Mechanics of systems with finitely many degrees of freedom, they are observables describing the particle positions. InQuantumFieldTheory, theobservablesarelocalquantitiesassociated each with a finite region in spacetime. They can never describe exactly a property of one particle or n - particle states, which are global (asymptotic) constructs. If that region reduces to a point, we find only the multiples of the identity. We ought to consider open regions. We might consider such a region as a neighbourhood of a spacetime point, defining it with some 4 uncertainty, and the measurement of associated local quantities as leading to information on that location. Thus Spacetime appears as a space of parameters, which, in absence of gravitational forces, can be specified with arbitrarily high (but finite!) precision,withhigherandhigherenergycostforhigherandhigherprecision. The consideration of the gravitational effects of that energy cost will cause, as we will see, that space of parameters to become noncommutative. The semiclassical level of a first analysis justifies also the use of concepts like energy; but a more careful analysis shows, as briefly mentioned here in thesequel, thatinessencetheconclusionsremaintruewithoutanyreference to the concept of energy. At a semiclassical level, the main consequence of the Principle stated above is the validity of Spacetime Uncertainty Relations; furthermore, they have been shown to be implemented by Commutation Relations between coordinates, thus turning Spacetime into Quantum Spacetime [1, 2]. The word “Quantum” is very appropriate here, to stress that noncommutativity does not enter just as a formal generalisation, but is strongly suggested by a compelling physical reason, unlike the very first discussions of possible noncommutativity of coordinates in the pre-renormalisation era, byHeisenberg,SnyderandYang,wherenoncommutativitywasregardedasa curious, in itself physically doubtful, possible regularisation device, without any reference to General Relativity and Gravitational forces; the qualita- tive fact that the quantum structure of gravitational forces ought to have consequences on the nature of spacetime in the small was anticipated by P.M.Bronstein [5], where, however, the focus was on the extension of the Bohr-Rosenfeld argument to the Christoffel symbols, and on the proposal of a Quantum Theory of linearised Gravity, without any mention of spacetime uncertainty relations. The analysis based on the Principle of Gravitational Stability against localisation of events leads to the following conclusions: i) There is no a priori lower limit on the precision in the measurement of any single coordinate (it is worthwhile to stress once more that the apparently opposite conclusions, still often reported in the literature inconnectionwiththeACVvariantoftheHeisenbergprinciple[6], are drawn under the implicit assumption that all the space coordinates of the event are simultaneously sharply measured). Every alerted reader will note that nobody knows an operational pre- scriptiontomeasure,say,only one spacetime coordinateofthelocation ofaneventwithaterrific(ultraPlanckian)precision. Butofcoursewe 5 cannot say that such a measurement is impossible just because we are not capable of inventing a device; we could say that only if we could show that it is forbidden by the presently known physical principles. Which at present does not seem to be the case. ii) The uncertainties ∆q in the measurement of the coordinates of an µ event in Minkowski space should be at least bounded by the following Spacetime Uncertainty Relations: 3 (cid:88) ∆q · ∆q (cid:38) 1; (1.2a) 0 j j=1 (cid:88) ∆q ∆q (cid:38) 1. (1.2b) j k 1≤j<k≤3 Thus points become fuzzy and locality looses any precise meaning. We believe it should be replaced at the Planck scale by an equally sharp and compelling principle, which reduces to locality at larger distances. Such a principle is nowadays totally unknown, and unaccessible by operational reasoning. Some comments on the derivation of these relations are in order. In the analysis of 1994–95, they were justified in special cases by their consis- tency with the exact solutions of Einstein Equations (EE), as Schwarzschild and Kerr’s solutions. But in general they were derived using the linearised approximation to EE. Furthermore the concept of energy was central: in a semiclassical ap- proach, the expectation value in a state describing an ansatz for the outcome of a localisation experiment (a coherent state in a free field theory) of the energy-momentum tensor for that field, was used as a source for the linearised EE. Then, the requirement of non-formation of trapped surfaces hiding the observed event was formulated as the condition of non negativity of the time-time component of the metric tensor. The relations above follow as a weaker simplified necessary condition. Both the use of the linearised approximation and of the notion of energy are doubtful. But in recent works [7, 8] Tomassini and Viaggiu have shown that (a stronger form of) the above relations do follow from an exact treatment, if one adopts the Hoop Conjecture, which limits the energy content of a space 6 volume in terms of the area of the boundary, as a condition for the non- formation of bounded trapped surfaces. Moreover, their analysis applies to a curved background as well. The treatment is again semiclassical, and involves the notion of energy, but the conflict about the use of the linearised approximations to derive bounds, and imposing those bounds in situations close to singularities, dis- appears. Eventually, in [4] the special case of spherically symmetric experiments, with all spacetime uncertainties taking the same value, was treated with use of the exact semiclassical EE, without any reference to the energy observables. The state describing the outcome of the localisation experiment was taken not as a strictly localised state, but as the state, with weaker localisation properties, obtained acting on the vacuum state with the field operators themselves, smeared with test functions having the appropriate symmetry, in a theory of a single scalar massless field coupled semiclassi- cally to gravity. The solution of the Raychaudhuri equation yields to the universal lower bound for the common value of the uncertainties, of the order of Planck length (see also Section 4.2 below for more details). We stress that this result gives a possibly weaker condition than the condition which could be derived by a choice of better localised ansätze for the probe state. We can conclude that the above Spacetime Uncertainty Relations are reasonably well grounded for Minkowski space; they are to be expected to hold in similar variant in curved spacetimes, by the Tomassini-Viaggiu argument; a basic consequence of those relations, when implemented by the Quantum Conditions we will now discuss, namely that the Planck scale is a universal minimal length, is well grounded on the basis of the most general assumptions, in the spherically symmetric case. The Spacetime Uncertainty Relations strongly suggest that spacetime has a Quantum Structure at small scales, expressed, in generic units, by [q ,q ] = iλ2Q , (1.3) µ ν P µν where Q has to be chosen not as a random toy mathematical model, but in such a way that (1.2) follows from (1.3). To achieve this in the simplest way, it suffices to select the model where the Q are central, and impose the “Quantum Conditions” on the two µν invariants Q Qµν; (1.4) µν 7   q ··· q 0 3 [q0,...,q3] ≡ det ... ... ...  q ··· q 0 3 ≡ εµνλρq q q q = µ ν λ ρ = −(1/2)Q (∗Q)µν; (1.5) µν whereby the first one must be zero and the square of the half of the second is I (in Planck units; we must take the square since it is a pseudoscalar and not a scalar). One obtains in this way [1, 2] a model of Quantum Spacetime which im- plements exactly our Spacetime Uncertainty Relations and is fully Poincaré covariant. As anticipated, here the classical Poincaré group acts as symmetries; translations, in particular, act adding to each q a real multiple of the iden- µ tity. Thus “coordinates” and “translation parameters”, classically described by the same objects, hear split into different entities; but this happens al- ready in non relativistic Quantum Mechanics: rotations apart, the Galilei group acts by adding numerical multiples of the identity to the non com- muting position and momentum operators . InviewoftheGel’fand–NaimarkTheorem,theclassicalMinkowskiSpace M is described by the commutative C*-algebra of continuous functions van- ishingatinfinityonM;theclassicalcoordinatescanbeviewedascommuting selfadjoint operators affiliated to that C*-algebras. SimilarlyanoncommutativeC*-algebraE ofQuantumSpacetimecanbe associated to the above relations. It was proposed in [1, 2] by a procedure which applies to more general cases (see also Sections 2.1 and 2.3 below). Assuming that the q ,Q are selfadjoint operators and that the Q λ µν µν commute strongly with one another and with the q , the relations above can λ be seen as a bundle of Lie algebra relations based on the joint spectrum of the Q . µν We are interested only in representations which are regular in the sense that in their central decomposition only integrable representations of the corresponding Lie algebras appear. Such representations are described by representations of the group C*- algebra of the unique simply connected Lie group associated to the corresponding Lie algebra. Hence the C*-algebra of Quantum Spacetime E is the C*-algebra of a continuous field of group C*-algebras based on the spectrum of a commuta- 8 tive C*-algebra. In our case, that spectrum—the joint spectrum of the Q —is the mani- µν foldΣoftherealvaluedantisymmetric2-tensorsfulfillingthesamerelations as the Q do: a homogeneous space of the proper orthochronous Lorentz µν group, identified with the coset space of SL(2,C) mod the subgroup of diagonal matrices. Each of those tensors can be taken to its rest frame, where the electric and magnetic part are parallel unit vectors, by a boost specified by a third vector, orthogonal to those unit vectors; thus Σ can be viewed as the tangent bundle to two copies of the unit sphere in 3-space—its base Σ . 1 The fibers, with the condition that I is not an independent generator but is represented by I, are the C*-algebras of the Heisenberg relations in 2 degrees of freedom—the algebra of all compact operators on a fixed infinite dimensional separable Hilbert space. The continuous field can be shown to be trivial, since it must contain a continuous field of one dimensional projectors—those corresponding to the orthogonal projection on the one dimensional subspace of multiples of the ground state vector for the harmonic oscillator (see [1]). ThestateswhosecentraldecompositionissupportedbythebaseΣ ,and 1 for each point of the base correspond to the ground state for the harmonic oscillator, are precisely the states of optimal localisation, where the sum of the four squared uncertainties of the coordinates is minimal, and equal to 2 (see Section 2.2 below). Thus the C*-algebra of Quantum Spacetime E is identified with the tensor product of the continuous functions vanishing at infinity on Σ and the algebra of compact operators. In the classical limit λ → 0 the second factor deforms to the commu- P tative C*-algebra of Minkowski space, but the first factor survives. When Quantum Spacetime is probed with optimally localised states its classical limit is M ×Σ , i.e. M acquires compact extra dimensions. 1 Note that the mathematical generalisation of points are pure states, but only optimally localised pure states are physically appropriate. But to explore more thoroughly the Quantum Geometry of Quantum Spacetime we must consider independent events. Quantum mechanically n independent events ought to be described by the n-fold tensor product of E with itself; considering arbitrary values on n we are led to use the direct sum over all n. If A is the C*-algebra with unit over C, obtained adding the unit to E, we will view the (n+1) tensor power Λ (A) of A over C as an A-bimodule n 9 with the product in A, and the direct sum ∞ (cid:77) Λ(A) = Λ (A) n n=0 as the A-bimodule tensor algebra, where (a ⊗a ⊗...⊗a )(b ⊗b ⊗...⊗b ) = a ⊗a ⊗...⊗(a b )⊗b ⊗...⊗b . 1 2 n 1 2 m 1 2 n 1 2 m This is the natural ambient for the universal differential calculus, where the differential is given by n (cid:88) d(a ⊗···⊗a ) = (−1)ka ⊗···⊗a ⊗I ⊗a ⊗···⊗a . 0 n 0 k−1 k n k=0 As usual d is a graded differential, i.e., if φ ∈ Λ(A),ψ ∈ Λ (A), we have n d2 = 0; d(φ·ψ) = (dφ)·ψ+(−1)nφ·dψ. Note that A = Λ (A) ⊂ Λ(A), and the d-stable subalgebra Ω(A) of Λ(A) 0 generated by A is the universal differential algebra. In other words, it is the subalgebra generated by A and da = I ⊗a−a⊗I as a varies in A. In the case of n independent events one is led to describe the spacetime coordinatesofthejth eventbyq = I⊗...I⊗⊗q⊗I...⊗I (q inthejth place); j in this way, the commutator between the different spacetime components of the q would depend on j. j A better choice is to require that it does not; this is achieved as follows. The centre Z of the multiplier algebra of E is the algebra of all bounded continuous functions on Σ with values in the complex numbers; so that E, and hence A, is in an obvious way a Z-bimodule. Therefore we can, and will, replace, in the definition of Λ(A), the C- tensor product by the Z-bimodule-tensor product, so that dQ = 0. As a consequence, the q and the 2−1/2(q − q ), j different from k, j j k and 2−1/2dq, obey the same spacetime commutation relations, as does the 10